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Existence of Compatible Systems of Lisse Sheaves on Arithmetic Schemes

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Existence of Compatible Systems of Lisse Sheaves on Arithmetic Schemes EXISTENCE OF COMPATIBLE SYSTEMS OF LISSE SHEAVES ON ARITHMETIC SCHEMES KOJI SHIMIZU Abstract. Deligne conjectured that a single `-adic lisse sheaf on a normal variety over a finite field can be embedded into a compatible system of `0-adic lisse sheaves with various `0. Drinfeld used Lafforgue’s result as an input and proved this conjecture when the variety is smooth. We consider an analogous existence problem for a regular flat scheme over Z and prove some cases using Lafforgue’s result and the work of Barnet-Lamb, Gee, Geraghty, and Taylor. 1. Introduction In [Del1], Deligne conjectured that all the Q`-sheaves on a variety over a finite field are mixed. A standard argument reduces this conjecture to the following one. Conjecture 1.1 (Deligne). Let p and ` be distinct primes. Let X be a connected normal scheme of finite type over Fp and E an irreducible lisse Q`-sheaf whose determinant has finite order. Then the following properties hold: (i) E is pure of weight 0. (ii) There exists a number field E ⊂ Q` such that the polynomial det(1 − Frobx t; Ex¯) has coefficients in E for every x 2 jXj. (iii) For every non-archimedean place λ of E prime to p, the roots of det(1 − Frobx t; Ex¯) are λ-adic units. (iv) For a sufficiently large E and for every non-archimedean place λ of E prime to p, there exists an Eλ-sheaf Eλ compatible with E, that is, det(1 − Frobx t; Ex¯) = det(1 − Frobx t; Eλ,x¯) for every x 2 jXj. Here jXj denotes the set of closed points of X and x¯ is a geometric point above x. The conjecture for curves is proved by L. Lafforgue in [Laf]. He also deals with parts (i) and (iii) in general by reducing them to the case of curves (see also [Del3]). Deligne proves part (ii) in [Del3], and Drinfeld proves part (iv) for smooth varieties in [Dri]. They both use Lafforgue’s results. We can consider similar questions for arbitrary schemes of finite type over Z[`−1]. This paper focuses on part (iv), namely the problem of embedding a single lisse Q`-sheaf into a compatible system of lisse sheaves. We have the following folklore conjecture in this direction. Conjecture 1.2. Let ` be a rational prime. Let X be an irreducible regular scheme that is flat and of finite type over Z[`−1]. Let E be a finite extension of Q and λ a prime of E above `. Let E be an irreducible lisse Eλ-sheaf on X and ρ the corresponding representation of π1(X). Assume the following conditions: (i) The polynomial det(1−Frobx t; Ex¯) has coefficients in E for every x 2 jXj. (ii) E is de Rham at ` (see below for the definition). 1 Then for each rational prime `0 and each prime λ0 of E above `0 there exists a lisse 0−1 Eλ0 -sheaf on X[` ] which is compatible with EjX[`0−1]. The conjecture when dim X = 1 is usually rephrased in terms of Galois repre- sentations of a number field (see Conjecture 1.3 of [Tay] for example). When dim X > 1, a lisse sheaf E is called de Rham at ` if for every closed point ∗ y 2 X ⊗ Q`, the representation iyρ of Gal k(y)=k(y) is de Rham, where iy is the morphism Spec k(y) ! X. Ruochuan Liu and Xinwen Zhu have shown that this is equivalent to the condition that the lisse Eλ-sheaf EjX⊗Q` on X ⊗ Q` is a de Rham sheaf in the sense of relative p-adic Hodge theory (see [LZ] for details). Now we discuss our main results. They concern Conjecture 1.2 for schemes over the ring of integers of a totally real or CM field. Theorem 1.3. Let ` be a rational prime and K a totally real field which is unram- −1 ified at `. Let X be an irreducible smooth OK [` ]-scheme such that • the generic fiber is geometrically irreducible, • XK (R) 6= ; for every real place K,! R, and • X extends to an irreducible smooth OK -scheme with nonempty fiber over each place of K above `. Let E be a finite extension of Q and λ a prime of E above `. Let E be a lisse Eλ-sheaf on X and ρ the corresponding representation of π1(X). Suppose that E satisfies the following assumptions: (i) The polynomial det(1−Frobx t; Ex¯) has coefficients in E for every x 2 jXj. (ii) For every totally real field L which is unramified at ` and every morphism ∗ α: Spec L ! X, the Eλ-representation α ρ of Gal(L=L) is crystalline at each prime v of L above `, and for each τ : L,! Eλ it has distinct τ-Hodge-Tate numbers in the range [0; ` − 2]. (iii) ρ can be equipped with symplectic (resp. orthogonal) structure with multi- × plier µ: π1(X) ! Eλ such that µjπ1(XK ) admits a factorization µK × µjπ1(XK ) : π1(XK ) −! Gal(K=K) −! Eλ with a totally odd (resp. totally even) character µK (see below for the definitions). (iv) The residual representation ρjπ1(X[ζ`]) is absolutely irreducible. Here ζ` is a primitive `-th root of unity and X[ζ`] = X ⊗OK OK [ζ`]. (v) ` ≥ 2(rank E + 1). Then for each rational prime `0 and each prime λ0 of E above `0 there exists a lisse 0−1 Eλ0 -sheaf on X[` ] which is compatible with EjX[`0−1]. For an Eλ-representation ρ: π1(X) ! GL(Vρ), a symplectic (resp. orthogonal) structure with multiplier is a pair (h ; i; µ) consisting of a symplectic (resp. orthog- × onal) pairing h ; i: Vρ × Vρ ! Eλ and a continuous homomorphism µ: π1(X) ! Eλ 0 0 0 satisfying hρ(g)v; ρ(g)v i = µ(g)hv; v i for any g 2 π1(X) and v; v 2 Vρ. We show a similar theorem without assuming that K is unramified at ` using the potential diagonalizability assumption. See Theorem 4.1 for this statement and Theorem 4.2 for the corresponding statement when K is CM. The proof of Theorem 1.3 uses Lafforgue’s work and the work of Barnet-Lamb, Gee, Geraghty, and Taylor (Theorem C of [BLGGT]). The latter work concerns Galois representations of a totally real field, and it can be regarded as a special 2 case of Conjecture 1.2 when dim X = 1. We remark that their theorem has several assumptions on Galois representations since they use potential automorphy. Hence Theorem 1.3 needs assumptions (ii)-(v) on lisse sheaves. The main part of this paper is devoted to constructing a compatible system of lisse sheaves on a scheme from those on curves. Our method is modeled after Drinfeld's result in [Dri], which we explain now. For a given lisse sheaf on a scheme, one can obtain a lisse sheaf on each curve on the scheme by restriction. Conversely, Drinfeld proves that a collection of lisse sheaves on curves on a regular scheme defines a lisse sheaf on the scheme if it satisfies some compatibility and tameness conditions (Theorem 2.5 of [Dri]). See also a remark after Theorem 1.4. This method originates from the work of Wiesend on higher dimensional class field theory ([Wie], [KS]). Drinfeld uses this method to reduce part (iv) of Conjecture 1.1 for smooth vari- eties to the case when dim X = 1, where he can use Lafforgue’s result. Similarly, one can use his result to reduce Conjecture 1.2 to the case when dim X = 1. However, Drinfeld's result cannot be used to reduce Theorem 1.3 to the results of Lafforgue and Barnet-Lamb, Gee, Geraghty, and Taylor since his theorem needs a lisse sheaf on every curve on the scheme as an input. On the other hand, the results of [Laf] and [BLGGT] only provide compatible systems of lisse sheaves on special types of curves on the scheme: curves over finite fields and totally real curves, that is, open subschemes of the spectrum of the ring of integers of a totally real field. Thus the goal of this paper is to deduce Theorem 1.3 using the existence of compatible systems of lisse sheaves on these types of curves. We now explain our method. Fix a prime ` and a finite extension Eλ of Q`. Fix a positive integer r. On a normal scheme X of finite type over Spec Z[`−1], each lisse Eλ-sheaf E of rank r defines a polynomial-valued map fE : jXj ! Eλ[t] of degree r given by fE;x(t) = det(1 − Frobx t; Ex¯). Here we say that a polynomial-valued map is of degree r if its values are polynomials of degree r. Moreover, fE determines E up to semisimplifications by the Chebotarev density theorem. Conversely, we can ask whether a polynomial-valued map f : jXj ! Eλ[t] of degree r arises from a lisse sheaf of rank r on X in this way. Let K be a totally real field. Let X be an irreducible smooth OK -scheme which has geometrically irreducible generic fiber and satisfies XK (R) 6= ; for every real place K,! R. In this situation, we show the following theorem. Theorem 1.4. A polynomial-valued map f of degree r on jXj arises from a lisse sheaf on X if and only if it satisfies the following conditions: (i) The restriction of f to each totally real curve arises from a lisse sheaf. (ii) The restriction of f to each separated smooth curve over a finite field arises from a lisse sheaf.
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